Parameter sensitivity of synthetic spectra and light curves of Type Ia supernovae

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Parameter sensitivity

of synthetic spectra and light curves

of Type Ia supernovae

Dissertation

zur Erlangung des Doktorgrades

an der Fakultät für Mathematik,

Informatik und Naturwissenschaften

Fachbereich Physik

der Universität Hamburg

vorgelegt von

Ernst Rolf Lexen

aus Kronstadt

Hamburg

2014

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Gutachter der Dissertation: Prof. Dr. Peter H. Hauschildt Prof. Dr. Edward A. Baron

Gutachter der Disputation: Prof. Dr. Robi Banerjee

Prof. Dr. Dieter Horns

Datum der Disputation: 24.01.2014

Vorsitzender des Prüfungsausschusses: Dr. Robert Baade

Vorsitzende des Promotionsausschusses: Prof. Dr. Daniela Pfannkuche

Dekan der MIN-Fakultät: Prof. Dr. Heinrich Graener

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Effulſiße autem Nouam aliquam Stellam, quæ a Mundi primordijs nuſquam antea patuerit, eandemque vltra integrum Annum in eodem Cœli loce perſeueraße, & ſucceßiue tandem diſparuiße, Miraculum eſt, omnium Hominum expectatione atque captu maius, & inter ea, quæ a Mundi primæua Origine in tota rerum Natura extiterunt, Literiſque prodita ſunt admi-randaſpectacula, ſi non maximum, ſaltem illis æquiparandum [...]1

(T. Brahe, 1602)

To strive, to seek, to find, and not to yield.

2

(A. Tennyson, 1853)

[...] the Universe is not only queerer than we suppose,

but queerer than we can suppose.3

(J. B. S. Haldane, 1927)

1_{Tichonis Brahe, 1602, in Astronomiae Instavratæ Progymnasmata [. . .], reprinted 1915, Tychonis Brahe}

Dani Opera Omnia, II, 317, 22-28

2_{Alfred Tennyson, 1853, in The poems of Alfred Tennyson 1830 - 1863, reprinted 1909,188}

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Kurzfassung

In dieser Arbeit werden die Parameter von Typ Ia Supernova Lichtkurven-Simulationen sys-tematisch variiert um die Sensitivität des Modells auf Parameteränderungen zu testen. Als Ausgangsmodell wird das parametrisierte Deflagrationsmodell W 7 verwendet. Die atmo-sphärische Struktur und die synthetischen Spektren werden mit PHOENIX berechnet. Zuerst wird der Anteil der radioaktiven Energie geändert. Dazu wird die initiale56Ni-Masse des W 7-Modells (m56_Ni ≡ 0, 568 m) in 10 %-Schritten von 50 % auf 150 %56Ni geändert.

Während die optische Tiefe sich dabei nicht wesentlich verändert, treten Temperaturunter-schiede von bis zu 11 · 103K auf. Die statistische Analyse der Lichtkurven ergibt keine Aus-reißer in den Bändern U, B und V (Johnson), u und g (SDSS), sowie Kp und D51 (Kepler). Für Modelle mit mehr56Ni verschiebt sich im Infraroten (IR) das zweite Maximum zeitlich weiter nach hinten. Dadurch kommt es teilweise zu einer Inversion der Helligkeiten in den Lichtkurven. Sie ist wellenlängenabhängig, was anhand der Bänder I (Johnson) und i (SDSS) gezeigt wird.

Danach wird die initiale Expansionsgeschwindigkeit (vmax_exp ≡ 30 · 103 _{km s}−1_{) in 10} %-Schritten zwischen 50 % und 150 % vexpvariiert. Dabei verschieben sich die optischen Tiefen (∆ logτ ' 1, 1) und es tritt eine Temperaturdifferenz von bis zu 33 · 103K auf. Modelle mit verringerter vexp sind heißer und erreichen ihr Maximum später.

Verändert man die beiden Parameter simultan, verstärken bzw. kompensieren sich diese Ef-fekte gegenseitig. Das lässt sich gut in den Spektren beobachten. In den Lichtkurven hinge-gen sind die Effekte nicht immer direkt ersichtlich.

Stabiles Eisen (Fe) wird in Abhängigkeit von der Geschwindigkeit der Schicht partiell durch 56_{Ni ersetzt. Zuerst wird dazu sämtliches Fe, das örtlich langsamer als}

. 4440 km s−1 ist, in Schritten von 20 % ersetzt. Das bedeutet insgesamt eine Erhöhung des56Ni um ' 12, 8 %. Die Auswirkungen sind geringer als bei einer globalen Erhöhung des56Ni-Anteils um 10 %. Bereits damit lässt sich die Phillips Relation auf eine differentielle Weise reproduzieren. Es kommt zu einer zeitlichen Verschiebung des zweiten Maximums und damit zu einer Inver-sion der Helligkeiten in den IR-Lichtkurven. In einem weiteren Schritt werden 50 % Fe für 7 Geschwindigkeiten an Orten_{. 4440 km s}−1durch56Ni ersetzt. Die Auswirkungen auf das Spektrum sind gering und können erst in den späten Spektren beobachtet werden.

Der Wert der parametrisierten Linienstreuung (εline) wird variiert, so dass zwischen 0 und 90 % der Photonen gestreut werden. Starke Streuung hat dabei große Auswirkung auf die Modellspektren, insbesondere im IR, wo die Effekte sehr deutlich zum Vorschein treten. Die Lage der Photosphäre ist wellenlängenabhängig: Im Ultravioletten (UV) verschiebt höhere Streuung die pseudo UV-Photosphäre näher an die Oberfläche und wir erhalten insgesamt eine höhere UV-Leuchtkraft. Im IR passiert das genaue Gegenteil.

Werden die drei Ionisationszustände von Calcium (Ca I, Ca II und Ca III) unter der Annahme von NLTE betrachtet, so ergeben sich deutliche Änderungen gegenüber den reinen LTE-Rechnungen in den Spektren und in den Lichtkurven. Die größten Abweichungen treten in den (Johnson) Bändern U, Ca II H und K, und I, Ca II (8579 Å) IR Triplett auf.

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Abstract

In this work, parameters of Type Ia supernova light curve simulations are systematically varied to test the model sensitivity due to parameter changes. As a starting model, the parametrized deflagration model W 7 is used. The atmospheric structure and the synthetic spectra are calculated with PHOENIX. Variations are done one at a time, first, the content of the radioactive energy is changed. For this purpose, the initial56Ni mass of the W 7 model (m56_Ni≡ 0.568 m) is changed in steps of 10 % from 50 % up to 150 %56Ni. While the

op-tical depth is not changed significantly, temperature differences occur up to 11 · 103K. The statistical analysis of the light curves shows no outliers in the bands U, B, and V (Johnson), u and g (SDSS), and Kp and D51 (Kepler). The second maximum in the infrared (IR) shifts further back in time for models with higher56Ni content. This leads to a partial inversion in the brightness of the light curves. The inversion is wavelength dependent and is prominent in the bands I (Johnson) and i (SDSS).

Then, the initial expansion velocity (vmax_exp ≡ 30 · 103km s−1) is varied in steps of 10 % from 50 % up to 150 % vexp. In this case, the total optical depths (∆ logτ ' 1.1) changes and a temperature difference of up to 33 · 103K occurs. Models with reduced vexp are hotter and reach maximum later.

Changing these two parameters simultaneously, they amplify or compensate each other. This can be well observed in the spectra. In the light curves, however, the effects are not always immediately apparent.

Stable iron (Fe) is partially replaced by56Ni, depending on the velocity of the layer. First, the entire Fe, which is at locations slower than_{. 4440 km s}−1, is successively replaced in steps of 20 %. This is a total increase of56Ni of 12.8 %. The effects are smaller than for the global increase of56Ni of 10 %. With this the Phillips relation can be reproduced differentially. It leads to a time shift of the second maximum, and thus an inversion of the brightness in the IR light curves. In a next step, 50 % Fe for 7 velocities at locations up to_{. 4440 km s}−1 are replaced by56Ni. The effects in the spectrum are small and can only be observed in the later spectra.

The value of the parametrized line scattering parameter, (εline) is varied so that between 0 and 90 % of the photons are scattered. Strong scattering has great impact on the model spectra, especially in the IR, where the effects are very clear. The position of the photosphere is wavelength dependent: In the ultraviolet (UV) higher scattering shifts the pseudo UV photosphere closer to the surface and we obtain a higher UV luminosity. In the IR, the exact opposite happens.

Considering the three ionization stages of calcium (Ca I, Ca II, and Ca III) under the assump-tion of non-local thermodynamic equilibrium (NLTE), significant changes compared to the pure LTE calculations can be observed, both in the spectra and in the light curves. The largest deviations occur in the (Johnson) bands U, Ca II H and K and I, Ca II (8579 Å) IR triplet.

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List of Figures

0.1 Supernova SN 2011by in the north of the center of NGC 3972 . . . iv

2.1 Spectra of SN 2011fe . . . 5

2.2 Light curves of SN 2011fe . . . 7

2.3 Sample of supernovae light curves from the Calán/Tololo Supernova Survey 10 2.4 Template light curves in the bands U, B, V, R, I, J, H, and K . . . 11

2.5 Hubble diagram for supernovae from the Calán/Tololo Supernova Survey and the Supernova Cosmology Project . . . 12

2.6 Bar graphs of the total number of discovered supernovae per year and cer-tainly recognized Type Ia supernovae since 1913 . . . 15

4.1 Geometric basis for the definition of the specific intensity . . . 22

4.2 Geometric volume element for the derivation of the radiative transfer equation 26 4.3 Evolution of the temperature as a function of the density of a white dwarf from the W 7 model . . . 33

4.4 Temperature and density at the deflagration front of the W 7 model as a func-tion of Lagrangian mass coordinate M/M . . . 34

4.5 Representation of the radioactive decay of56Ni and56Co . . . 38

4.6 Time evolving series of spectra . . . 44

4.7 Johnson filter functions U, B, V, R, and I . . . 45

4.8 Johnson filter functions J, H, and K . . . 45

4.9 Filter functions for the photometric systems 2MASS, UKIDSS, SDSSS, and Kepler . . . 46

4.10 Photometric systems and the wavelength range of its passbands . . . 48

4.11 Iteration scheme of PHOENIX . . . 49

4.12 Model light curves for the fiducial model for U, B, V, R, and I (for the John-son photometric system) and for J, H, K (2MASS) . . . 51

4.13 Same as in Figure 4.12 but for Z, Y, J, H, and K (for UKIDSS) and for Kp (Kepler) . . . 52

4.14 Same as in Figures 4.12 and 4.13 but for u, g, r, i, and z (SDSS) and D51 (Kepler) . . . 53

5.1 Normalized initial composition structure of W 7 . . . 55

5.2 Entire synthetic spectrum, as well as a section . . . 56

5.3 Particles of 56Ni as a function of speed for models in the range with 50 % reduced56Ni and 50 % enriched56Ni . . . 58

5.4 Synthetic spectra for day 20 after the explosion for varying56Ni . . . 59

5.5 Normalized particle number of ionized iron as a function of the optical depth τ and plotted against temperature for day 20 after the explosion . . . 60

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List of Figures

5.6 Same as in Figure 5.5, but for day 40 after the explosion . . . 61 5.7 Synthetic spectra for days 5, 10, 14, 16, 18, 20, 22, 24, 26, 30, 40, and 50 of

models for varying56Ni abundances . . . 63 5.8 Same as in Figure 5.7, but for the wavelength range of the B band . . . 64 5.9 Spatial representation of model light curves in the bands U, B, V, R, I, J, H,

and K for varying56Ni . . . 65 5.10 Model light curves in the bands U, B, V, R, I, J, H, and K for varying56Ni . 66 5.11 Summarized graphical representation of the statistical analysis of the model

light curves from Figure 5.9 in the bands U, B, V, and R. . . 68 5.12 Summarized graphical representation of the statistical analysis of the model

light curves from Figure 5.9 in the bands I, J, H, and K. . . 69 5.13 Pearson correlation coefficients for the individual days of the light curve of

the B band for models with varying56Ni . . . 70 5.14 Summed absolute values of the differences between two adjacent models for

previously discussed bands shown in Figures 5.9 and 5.10 . . . 71 5.15 Summarized graphical representation of the statistical analysis of the model

light curves from Section 4.5, Figure 4.12 for the Johnson photometric sys-tem for the bands U, B, V, R, and I and for 2MASS for the bands J, H, and K 73 5.16 Summarized graphical representation of the statistical analysis of the model

light curves from Section 4.5, Figure 4.13 for the UKIDSS photometric sys-tem for the bands Z, Y, J, H, and K and for Kepler for the band Kp . . . 74 5.17 Summarized graphical representation of the statistical analysis of the model

light curves from Section 4.5, Figure 4.14 for the SDSS photometric system for the bands u, g, r, i, and z and for Kepler for the band D51 . . . 75 5.18 Temperature stratifications as a function of the optical depth for the 32

mod-els from which the light curve was calculated for the fiducial model . . . . 77 5.19 Temperature stratifications as a function of the optical depth for the 32

mod-els from which the light curve was calculated for the modmod-els with 50 %56Ni and 150 %56Ni . . . 78 5.20 Temperature profiles in 103K as a function of the optical depth for models

with varying56Ni abundance for days 5, 10, 14, 16, 18, 20, 22, 24, 26, 30, 40, and 50 . . . 80 5.21 Discretized velocity model profiles as a function of the layers . . . 82 5.22 Synthetic spectra for day 20 after the explosion for varying vexp . . . 83 5.23 Synthetic spectra for days 5, 10, 14, 16, 18, 20, 22, 24, 26, 30, 40, and 50 of

models for varying vexp . . . 84 5.24 Spatial representation of model light curves in the bands U, B, V, R, I, J, H,

and K for varying vexp . . . 85 5.25 Model light curves in the bands U, B, V, R, I, J, H, and K for varying vexp . 86 5.26 Summed absolute values of the differences between two adjacent models for

bands of models shown in Figures 5.24 and 5.25. . . 87 5.27 Temperature stratifications as a function of the optical depth for the 32

mod-els from which the light curve was calculated for the modmod-els with 50 % vexp and 150 % vexp . . . 89

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List of Figures

5.28 Synthetic spectra for the day 20 after the explosion for varying simultane-ously56Ni and vexp . . . 91 5.29 Model light curves in the bands U, B, V, R, I, J, H, and K for varying

simul-taneously56Ni and vexp . . . 92 5.30 Mass fractions of Fe and 56Ni as a function of velocity at locations up to

4440 km s−1 . . . 94 5.31 Normalized (to the mass of the sun) mass fractions of Fe and56Ni as a

func-tion of velocity . . . 95 5.32 Course of the velocity dependent substitution of Fe by56Ni . . . 95 5.33 Synthetic spectra for days 5, 20, and 50 after the explosion for replaced Fe

in the core. The corresponding models are shown in Figures 5.30 and 5.32 96 5.34 Synthetic spectra for day 20 after the explosion for Fe replaced in the core . 97 5.35 Model light curves in the bands U, B, V, R, I, J, H, and K. Fe was partially

replaced by56Ni at locations up to_{. 4440 km s}−1 . . . 98 5.36 Synthetic spectra for day 20 after the explosion for varying εline . . . 100 5.37 Synthetic spectra for days 5, 10, 14, 16, 18, 20, 22, 24, 26, 30, 40, and 50 of

models for varying εline . . . 101 5.38 Spatial representation of model light curves in the bands U, B, V, R, I, J, H,

and K for varying εline . . . 103 5.39 Model light curves in the bands U, B, V, R, I, J, H, and K for varying εline . 104 5.40 The summed absolute values of the differences of the models for different

εlineshown in Figure 5.38. . . 105 5.41 Temperature stratifications as a function of the optical depth for the 32

mod-els from which the light curve was calculated for the modmod-els with εline= 0.1 and εline= 1 . . . 106 5.42 Synthetic NLTE spectra for day 20 after the explosion . . . 108 5.43 NLTE model light curves in the bands U, B, V, R, I, J, H, and K . . . 109 5.44 Synthetic NLTE spectra within the wavelength region of the I band for day 20

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List of Tables

4.1 Compilation of passbands used in this work, specifying the filter name, the effective wavelength midpoint, the bandwidth, the specifying region, and the description of the photometric system. . . 47 5.1 Summarized results for the temperatures and the range of the optical depths

for days 5, 20, and 50 for the fiducial model (100 % 56Ni) shown in Fig-ure 5.18 and for the two extreme models with 50 % 56Ni and 150 % 56Ni shown in Figure 5.19 . . . 79 5.2 Summarized results for the temperatures and the range of the optical depths

for days 5, 20, and 50 for the fiducial model (100 % vexp) shown in Fig-ure 5.18 and for the two extreme models with 50 % vexp and 150 % vexp shown in Figure 5.27 . . . 88 5.3 Summarized results for the temperatures and the range of the optical depths

for days 5, 20, and 50 for the fiducial model (εline = 0.8) shown in Fig-ure 5.18 and for the two extreme models with ε_line = 0.1 and εline = 1.0 shown in Figure 5.41 . . . 106

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Chapter 1 Introduction

Type Ia supernovae are among the brightest single objects in the Universe. It is generally agreed that they arise from the thermonuclear explosion of a white dwarf in a binary system. Due to the high luminosity of these objects and their explosion near a critical mass limit they became important cosmological distance indicators.

A general overview about supernovae is given in Section 2 of this work. We start with a historical overview (Section 2.1), and have a closer look at empirical classification and diversity, based on spectra and light curves (Section 2.2). Thereafter, an overview about progenitors and the physical background of Type Ia supernovae is given (Section 2.3). In Section 2.4 we focus on the standardizability of spectroscopically normal and canonical Type Ia supernovae light curves. Finally, in Section 2.5 definite departure from the canonical behavior based on observational data is shown. The strong increase of observational data since the 1990s (cf. Section 2.5) showed that Type Ia supernovae are not standard candles, but a major part can be transformed in standardizable candles.

However, intrinsic brightness differences are caused by different effects. Nevertheless, the unusual light curve decline of some peculiar supernovae can be explained within the frame-work of normal supernovae. The objects can be modeled under the assumption of extended individual parameters (cf. Baron et al. (2012) and Section 2.5).

The state of the art method to determine model parameters is parameter estimation based on mathematical methods as proposed by Bock (1981, 1987). These methods are presented in Section 3 especially for the astrophysical problem of parameter estimation. The individual parameters and the parameter sensitivity of Type Ia supernovae are determined.

However, these kinds of inverse problems can only be applied to single objects, where models and parameters are individually adjusted.

The aim of this work is to analyze the behavior of simulation results, e.g., spectra and light curves in different passbands, to change the underlying explosion model by varying the struc-tures systematically. This is important as it helps us to quantify the uncertainties of parameter determinations through light curve simulations. Thus naturally depends on a series of related problems, e.g., requests for parameters which are poorly constrained by the observations and the modeling approach and that can, therefore, not be measured reliably. Additional ques-tions are connected, e.g., effects on the model by changing technical or algorithmic parame-ters.

It is clear, therefore, that the systematic investigation of these dependencies considered as classical forward problem is truly astronomical. On the other hand, a fundamental obstacle is the fact that we cannot simply "reverse" the simulations based on the solution of integro-differential equations (here: the time dependent radiative transfer equation with scattering),

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Chapter 1 Introduction

while linear, cannot be directly reversed.

Therefore, we will use a methodology that instruments the forward simulations with PHOENIX/1D to provide information about the reaction of the simulation results, e.g., syn-thetic spectra and light curves, to vary the different sets of input parameters directly.

Following, the results will be discussed and the conclusions and the outlook (Section 6) show the necessity of a quantitative understanding of this extensive and complex problem.

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Chapter 2 Supernovae

2.1 Historical approach

As one of the brightest events in the Universe, supernovae have great importance and formed, both in the distant and in the recent past, milestones in our understanding of stellar evolution and cosmology. In Europe, the first "new star" (nowadays we call such stars supernovae) was discovered by Tycho Brahe in the constellation Cassiopeia in 15721. He wrote about it "De nova et nullius aevi memoria prius visa stella, [. . .]"2. As the new discovered star did not change its position against the fixed stars, Brahe concluded that the star had to belong to the fixed stars. Thus began a new era, away from Aristotle and dogmatic belief of the Church of the immutability of the heavens. Previously, there had already been more observations of so-called "new stars", such as SN 1054, now known as the Crab Nebula3, which was only observed outside Europe, but they were simply not known or ignored by European scholars. Clark & Stephenson (1977), Green & Stephenson (2003), and Stephenson & Green (2005) summarized the historical "evolution" of supernovae.

Astronomically a split second later, followed the discovery of the second, and until now, the last supernova in our galaxy in 1604. It was Kepler’s supernova, in the constellation Ophiuchius4. He wrote about it "De stella nova in pede Serpentari, [. . .]"5.

The rate in which Type Ia supernovae occur in a galaxy like the Milky Way is only a few per millennium. Supernova rates and estimates are discussed in, e.g., Cappellaro et al. (1997) and the Type Ia supernovae rates especially in, e.g., Pain et al. (1996, 2002) and Graur & Maoz (2013). Supernova rates are briefly discussed in Section 2.5, where they are shown graphically in Figure 2.6.

But what are supernovae? The term was introduced by Walter Bade and Fritz Zwicky (cf. Osterbrock 2001).

Commonly, supernovae are thought to be exploding stars whose luminosity increase abruptly by several orders of magnitude and cause a burst of radiation that often briefly outshines the entire host galaxy.

1_{It was the supernova SN 1572}

2_{Tychonis Brahe, Dani, 1573, in De nova et nullius aevi memoria prius visa stella, iam pridem anno à nato}

Christo 1572, mense Novembri primum conspecta, contemplatio mathematica [. . .]

3_{M1 in the Messier Catalogue}

4_{Formerly known as Serpentarius}

5_{Joannis Keppleri, 1606, De Stella Nova in Pede Serpentarii, et qui sub ejus Exortum de novo iniit, Trigono}

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Chapter 2 Supernovae

2.2 Empirical classification and diversity based on spectra

and light curves

It took until 1885 until the first spectrum of a supernova was published6 (Sherman 1885, 1886, and for a summary Gaposchkin (1936)) and almost another 60 years until Popper (1937) noted that there may be more kinds of supernovae.

The two main classes, the Types I and II were "provisionally" established by Minkowski (1941), after he had deviated from his earlier position that all supernovae have similar spec-tra and only differ in details of minor importance (Minkowski 1939). Of his 14 supernovae, nine spectra were extremely homogeneous, while the remaining five were distinctly dif-ferent. He took the division and could show that Type I supernovae have, compared to those of Type II, lower expansion velocities and no lines of hydrogen or any other elements (Minkowski 1941). At this time it was very difficult to obtain good spectra of supernovae and some had to be withdrawn because they turned out to be not spectra of supernovae, e.g., Strohmeier (1938). This was caused by the instruments, the absence of linear detectors, but also the very large distances of these objects. Therefore, photometry was mainly used to obtain light curves. From changes in luminosity with time, which is measured as the evolu-tion of the apparent brightness, conclusions can be drawn on the properties of the observed object. Zwicky (1938) and Baade & Zwicky (1938) realized the importance of the light curves for the understanding of supernovae. Today, they are, still, an important tool and also the historical observations can be used today. Branch (1990) gave a historical overview of spectra and Kirshner (1990) of light curves of supernovae.

Despite some additions and corrections to the current taxonomy, such as the subclassification of the Type I supernovae (Bertola 1962) into classical Ia (cf. Elias et al. 1985; Harkness & Wheeler 1990; Wheeler & Harkness 1990; Filippenko 1997; Wheeler & Benetti 2000) and the extension to the physically distinct class Ib, which is characterized by the presence of neutral helium He I (Harkness et al. 1987) and the absence of the strong singly ionized sili-con Si II absorption feature at 6150 Å (Pskovskii 1969; Branch & Patchett 1973), and later the branch into another variety with poor He I, Ic (Elias et al. 1985; Wheeler & Levreault 1985; Porter & Filippenko 1987; Harkness & Wheeler 1990; Wheeler & Harkness 1990; Filippenko 1997; Wheeler & Benetti 2000), or the rearrangement of Zwicky’s (1965) newly introduced Types III, IV, and V supernovae among Type II supernovae, the present day clas-sification scheme is useful to sort out supernovae according to their characteristics and to look for similarities and differences. Regardless of the achievements in the classification, it is ambiguous and the many peculiar objects make it even more confusing. For example, there are peculiar types as the super-Chandrasekhar mass Type Ia supernovae, or a distinct class within the peculiar supernovae, the Type Iax (Foley et al. 2013).

A review of the supernovae taxonomy is given in, e.g., Harkness & Wheeler (1990), da Silva (1993), Filippenko (1997), and Wheeler & Benetti (2000).

Type Ia supernovae are traditionally defined by their spectra, which contain much informa-tion about the ejected matter, such as the composiinforma-tion and the expansion velocities. One of its main features is the total absence of hydrogen in their spectra near maximum light (Minkowski 1941).

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2.2 Empirical classification and diversity based on spectra and light curves

0 . 0 0 E + 0 0 0

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Figure 2.1: Spectral evolution of the Type Ia supernova SN 2011fe from 08-28-2011 (top) to 10-21-2011 (bottom). In the upper right corner of each spectrum is the Modified Julian Date specified. In the early spectra Ca II, Mg II, Fe II, Si II were identified (Parrent et al. 2012). See text for details.

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A second characteristic is the empirical subdivision based on features in their spectra, such as the presence of the strong singly ionized silicon (Si II) absorption feature at 6150 Å, produced by blueshifted Si II (6347 Å and 6371 Å) (Pskovskii 1969; Branch & Patchett 1973). Together with the two Fraunhofer lines of singly ionized calcium Ca II, H and K (3968.5 Å and 3933.7 Å) they are the strongest lines in the spectrum. Other prominent lines near maximum light are singly ionized silicon Si II (3858 Å, 4130 Å, 5051 Å, and 5972 Å), singly ionized calcium Ca II (8579 Å), singly ionized magnesium Mg II (4481 Å), singly ionized sulfur S II (5468 Å, 5612 Å, and 5654 Å), and neutral oxygen O I (7773 Å). These features are usually formed by resonant scattering above the photosphere and show strong P Cygni profiles characteristic of expanding atmospheres (Filippenko 1997).

There are some contributions from lowly ionized iron and iron peak elements, which increase after maximum brightness (Mazzali et al. 1997).

In Figure 2.1 the spectra of the supernova PTF11kly/SN 2011fe7 are shown in the wave-length range [3000, 10000] Å starting from 08-28-2011 to 10-21-2011. In particular, the days specified in Modified Julian Date (MJD) are: 55801.12 (08-28-2011), 55804.25 (08-31-2011), 55807.43 (09-03-(08-31-2011), 55811.42 (09-07-(08-31-2011), 55814.43 (09-10-(08-31-2011), 55817.43 (09-13-2011), 55823.61 (09-19-2011), 55835.26 (10-01-2011), 55841.31 (10-07-2011), and 55855.17 (10-21-2011).

In Figure 2.2 the light curves in the bands B, V, R, and I of the supernova PTF11kly/SN 2011f are shown for the first 350 days.

SN 2011fe was detected by the Palomar Transient Factory (Law et al. 2009; Rau et al. 2009) in the Pinwheel galaxy8, a spiral galaxy at a distance of 6.4 Mpc (Shappee & Stanek 2011), the closest9 Type Ia supernova in the past 25 years (Nugent et al. 2011). The onset of the event was established based on models at UT 2011, August 23, 16:29 and Ca II, Mg II, Fe II, Si II were identified in the early spectra (Nugent et al. 2011).

Filippenko (1997) urged caution of uncorrected spectra. The spectra may be contaminated with hydrogen from superimposed H II regions or contain emission lines from circumstellar material. A discussion of the presence or absence of circumstellar material is given in, e.g., Branch et al. (1995). Type Ia supernovae are usually not associated with star formation regions (and, therefore, with H II regions). However, there are exceptions, such as SN 2002ic (Wood-Vasey et al. 2002a; Hamuy et al. 2002), the first observed Type Ia supernova with strong Hα and a weaker Hβ emission (Hamuy et al. 2003a) caused by the interaction with hydrogen-rich circumstellar medium (Hamuy et al. 2003b; Wang et al. 2004). However, it is also possible, based on the light curve behavior and the Hα profile, that it is a Type IIn supernova, whereas it was spectroscopically identified as Type Ia (Hamuy et al. 2002, 2003b; Wood-Vasey et al. 2004).

Unlike other supernovae, Type Ia supernovae occur in all morphological types of galaxies and are usually associated with older stellar populations. In spiral galaxies they are concen-trated in spiral arms only to the extent where the oldest stars are (Maza & van den Bergh 1976; McMillan & Ciardullo 1996; Wheeler & Benetti 2000). It is now known that there is a relation between the luminosity of Type Ia supernovae and the morphological

classifi-7_{The latter is the official name of the IAU}

8_{M101 in the Messier Catalogue}

9_{Before SN 2011fe a closer Type I supernova was SN 1986G (Evans et al. 1986) in M83 (specifically}

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2.2 Empirical classification and diversity based on spectra and light curves 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 2 0 1 8 1 6 1 4 1 2 1 0 m b o l + c o n s t. T i m e [ D a y s ] B V R I

Figure 2.2: Light curves of the Type Ia supernova SN 2011fe in the bands B, V, R, and I. The data for each passband have been offset vertically (Baron et al. forthcoming).

cation of the host galaxy. Hamuy et al. (1995, 1996b) found that the intrinsically brightest Type Ia supernovae occur in late-type galaxies. Likewise, it was found by Howell (2001) that peculiar subluminous Type Ia supernovae come from an old population. Overall, it can be said that the brightness and the width of the light curve are correlated with the mass, the star formation rate, the age, and the metallicity of the host galaxy.

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2.3 Progenitors and physical background of Type Ia

supernovae

It is thought that Type Ia supernovae are carbon-oxygen white dwarfs that exceed by ac-cretion of mass from a companion in a binary system a critical mass limit (Chandrasekhar 1931, 1939). It was named after its discoverer ChandrasekharMChan10. As soon as the white dwarf grows (close) toMChan, carbon is ignited. In the ensuing explosion the white dwarf will completely destroyed (Hoyle & Fowler 1960). The nature of the accretion process and the companion still remain open questions. One of the possibilities is the single-degenerate scenario, where the companion star is a main sequence star or a red giant (Whelan & Iben 1973). In the double-degenerate scenario the companion is another white dwarf (Iben & Tutukov 1984; Webbink 1984). Another option is sub-Chandra, where a helium layer ac-cumulates on a carbon-oxygen white dwarf (Woosley & Weaver 1986). When the pressure exceeds a critical threshold value, helium is ignited and the white dwarf (markedly) below the Chandrasekhar mass detonates due to the inward propagating shock front compressing the carbon-oxygen core which ignites (Nomoto 1980). Other compositions of the white dwarf, such as oxygen-neon-magnesium or helium are possible, but can be excluded for this sce-nario, because they play only a minor role due to physical and statistical considerations11. The identification of the progenitor system remains a major unsolved problem (Maoz & Mannucci 2012). Even the very early discovery of the "nearby" supernova SN 2011fe in the era of modern instruments could not resolve this question, but some scenarios, such as the companion star being a red giant, could be excluded (Nugent et al. 2011; Li et al. 2011; Horesh et al. 2012). Bloom et al. (2012) determined the radius of the progenitor and con-cluded that it is a compact object. Nugent et al. (2011) concon-cluded from the lack of an early shock that the companion was most likely a main sequence star. Using archive images of the region from the Hubble Space Telescope, Li et al. (2011) ruled out almost all helium stars as companion stars12. Using radio data, Chomiuk et al. (2012) placed constraints on the density of medium the progenitor surrounding and interpreted it as ruling out the single degenerate scenario, even limited the double degenerate scenario and did not rule out "exotic" scenarios or the core degenerate scenario.

The explosion of a carbon-oxygen white dwarf itself is the result of a thermonuclear runaway in degenerate carbon burning. This process is poorly understood theoretically, despite some progress. In particular, the conditions of the "simmering", the exact moment until nuclear fusion begins, where it starts, and if it starts possibly several times and ultimately the mech-anism that triggers the detonation is vague. It is commonly thought that the nuclear burning "flame" front propagates initially at subsonic speed as a deflagration wave outward. Fusion processes take place and many of the (neutral and singly ionized) intermediate mass elements lighter than iron, such as oxygen, magnesium, silicon, sulfur, and calcium are produced. A closer look on the nucleosynthesis is given in, e.g., Hoyle & Fowler (1960, 1961), Truran et al. (1967), Thielemann et al. (1986), and Travaglio & Hix (2013). At different

densi-10_M

Chan= 1.46 0.5Ye

2

M, where Yeis the electron mole fraction (Chandrasekhar 1939)

11_{An oxygen-neon-magnesium white dwarf will not end as a Type Ia supernova (cf. Miyaji et al. 1980)}

12_{"This directly rules out luminous red giants and the vast majority of helium stars as the mass-donating}

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2.3 Progenitors and physical background of Type Ia supernovae

ties, the flame front becomes Taylor unstable (cf. Chandrasekhar 1961). Rayleigh-Taylor instabilities enlarge and deform the surface of the flame front, as also known from the chemical combustion physics, e.g., Jäger et al. (2007). Turbulence plays a major role in the mixing of unburnt material in the deformed burning front. In the outer layers, ultimately a detonation wave is induced by fast deflagration (cf. Khokhlov et al. 1993; Nomoto et al. 1996). The transition in the unregulated carbon burning from deflagration (subsonic burning wave) to detonation (supersonic) is, still, not well understood but is, in fact, needed, since a pure deflagration would not provide enough power and the flame would fizzle, whereas a pure detonation would lead to an overabundance of iron group elements and not enough of the intermediate mass elements observed in the peak phase. Ultimately, the flame front is accelerated to supersonic speed, and the released nuclear energy is ripping the white dwarf completely apart. It releases a kinetic energy of ≈ 1051erg (Khokhlov et al. 1993) often ab-breviated "foe" ([ten to the power of] fifty-one ergs)13. Here all burnt and unburnt material is ejected into space. Partial trapping and thermalization of β -unstable56₂₈Ni (half-life 6.075 d (Junde et al. 2011)) and its daughter nucleus56₂₇Co (half-life 77.236 d (Junde et al. 2011)) play an important role. Both isotopes power the heating of the adiabatically cooling shell through gamma ray emission,

56 28Ni28 e−capture −−−−−−−→ T1/2=6.075 d 56 27Co29 e−capture & β+ −−−−−−−−→ T1/2=77.236 d 56 26Fe30.

While56₂₈Ni decays only by electron capture, 56₂₇Co decays by both electron capture (81 %) and β+ decay (19 %) (Leibundgut 2000).

The idea of nuclear heating through decaying isotopes goes back to Truran et al. (1967) and Colgate & McKee (1969) and has been confirmed by measurements of Kuchner et al. (1994).

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2.4 Standardizability and cosmological applications

Due to the huge luminosity, it becomes possible to see Type Ia supernovae far away in space and, therefore, far back in time. Thus, when the white dwarf reaches the same critical mass, MChan, before it is fully disrupted by the thermonuclear explosion, it is believed that the explosion energy is nearly the same. All types of supernovae, except for the Type Ia, are very inhomogeneous and can ultimately not be used for inductive statements14. Canonical Type Ia supernovae light curves look remarkably similar. Most of them differ only slightly. Especially in the B and V bands, where the majority of the energy of Type Ia supernova is emitted, there are only small deviations around a template behavior.

Figure 2.3: In the upper panel a sample of light curves from low redshift Type Ia supernovae from the Calán/Tololo Supernova Survey (Hamuy et al. 1996a) is shown. The brightness in the B band as a function of time is shown before and after maximum (day 0). The light curves of brighter supernovae decline more slowly whereas the decrease of the dimmer is faster (Phillips 1993). In the lower panel the same light curves are shown after calibration. Corrected is the stretch of the timescale and the intrinsic peak of the color. The figure is based on Perlmutter (1999) (and references therein), with modifications.

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2.4 Standardizability and cosmological applications

This could be shown with the measured Type Ia supernovae from the Calán/Tololo Super-nova Survey (Hamuy et al. 1993, 1996a). Light curves of this superSuper-nova survey are shown in Figure 2.3, where the Phillips relation (Phillips 1993) can be seen very well. Phillips showed a linear relationship between a defined parameter, ∆m15, which describes the de-crease of the B magnitude within the first 15 days after the maximum in the B band and the absolute peak luminosity. The light curves of brighter supernovae decline more slowly, whereas the decrease of the dimmer is faster. These results based on the investigation from the Calán/Tololo Supernova Survey and were earlier proposed by Barbon et al. (1973) and Pskovskii (1977, 1984). In the lower panel of Figure 2.3, the standardized profile of the re-calibrated light curves is shown. These data were essential to show that supernovae may be standard candles and also provide an important basis for subsequent work. Later, the Phillips relation was further developed and improved (Hamuy et al. 1995, 1996a; Phillips et al. 1999). Other methods based on the initial idea are the multicolor light-curve shape method (Riess et al. 1995, 1996, 1998; Jha et al. 2007) and the stretch correction method (Perlmutter et al. 1997; Goldhaber et al. 2001).

At least since the discovery and observation of supernovae in galaxies with known dis-tances it becomes possible to use Type Ia supernovae as distance indicators in cosmology. Cepheids15 are used to determine distances (Hubble 1925). However, they are only suitable for (astronomical) nearby galaxies. With the appearance of supernovae in such galaxies16, direct comparison methods were thus available (Panagia et al. 1992; Sandage et al. 1992, 1994, 1996; Saha et al. 1995, 1996a,b, 1997, 1999, 2001a,b).

- 2 0 0 2 0 4 0 6 0 6 5 4 3 2 1 M a g n it u d e + C o n s t. T i m e [ d a y s ] K H + 0 . 5 I + 1 . 5 J + 1 . 8 - 2 0 0 2 0 4 0 6 0 5 4 3 2 1 0 M a g n it u d e + C o n s t. T i m e [ d a y s ] R V + 0 . 3 B + 0 . 9 U + 1 . 5

Figure 2.4: Plotted are the magnitudes as a function of time of the template light curves in the bands U + 1.5, B + 0.9, V + 0.3, and R (left panel) and I + 1.5, J + 1.8, H + 0.5, and K (right panel). The legend labels the bands and the added constants. The figure is based on the data of Nugent et al. (2002), with modifications. See text for details.

15_{After the eponymous star Delta Cephei (for classical Cepheids). There is a relationship between their periods}

and luminosities (Leavitt & Pickering 1912).

16_{Recently SN 1895B (Johnson 1936; Hoffleit 1939) and SN 1937C (Baade & Zwicky 1938; Minkowski 1939)}

are the first known Type Ia supernovae whose parent galaxies, NGC 5253 and IC 4182, has been reached by Cepheids (Panagia et al. 1992; Sandage et al. 1992, 1994; Saha et al. 1995)

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Wilson (1939) suggested as one of the first to use supernovae for the study of nebular red-shifts and thus for cosmological distances. It was followed by contributions to the determina-tion of cosmological parameters, e.g., Branch & Patchett (1972), Branch (1977a,b, 1985a), Branch & Tammann (1992), Riess et al. (1998), and Perlmutter et al. (1999). In 1992, Branch & Tammann discussed Type Ia supernovae as standard candles and concluded that with ad-vances in observation and modeling, they are certain to become increasingly valuable as extragalactic distance indicators. Branch et al. (1993) defined from an observational sample spectroscopically normal and peculiar Type Ia supernovae. Ultimately, out of this a tem-plate has been developed whose supernovae are commonly called "Branch-normal Type Ia supernovae". Nugent et al. (2002) showed a further development of this scheme for the near-infrared bands. Their data for the template light curves in the bands U, B, V, R, I, J, H, and K are shown in Figure 2.4.

Figure 2.5: The Hubble diagram for 18 low redshift Type Ia supernovae from the Calán/Tololo Supernova Survey and 42 high redshift supernovae from the Super-nova Cosmology Project, plotted on a linear redshift scale. The solid curves rep-resent a range of cosmological models with Λ = 0, Ω_Λ= 0, and ΩM= 0, 1, and 2. The dashed curves show a range of flat cosmological models for ΩM = 0, 0.5, 1, and 1.5 where ΩM+ ΩΛ = 1. The unfilled circles indicate not included super-novae. The figure is from Perlmutter et al. (1999), with modifications.

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2.4 Standardizability and cosmological applications

They used the supernovae SN 1981B, SN 1989B, SN 1990N, SN 1990af, SN 1992A, SN 1992P, SN 1992ag, SN 1992al, SN 1992aq, SN 1992bc, SN 1992bh, SN 1992bl, SN 1992bo, SN 1992bp, SN 1992br, SN 1992bs, SN 1992b, SN 1992b, SN 1992b, SN 1993H, SN 1993O, and SN 1994D to create this template.

The near infrared has many advantages, but it is not easy to get good spectra in this region. After all, Marion et al. (2003), still, had claimed that with the release of twelve near infrared spectra, they more than doubled the number of published spectra within three weeks of max-imum light. Marion et al. (2009) published 41 near infrared spectra, however, which meant a huge improvement to the relatively few published in the literature at this time.

The first high redshift Type Ia supernovae was observed at z = 0.31 (Norgaard-Nielsen et al. 1989), the most distant one at z = 1.914 (Jones et al. 2013), corresponding to a lookback time of the order of& 10 Gy. The most important discovery was that the expansion of the Universe is not slowing down, due to the attractive force of gravity, but is, instead, accelerated (Riess et al. 1998; Perlmutter et al. 1999). The force responsible is called the vacuum energy, often "dark energy", and constitutes ≈ 70 % (ΩΛ= 0.713) of the content of the Universe (Kowalski et al. 2008). Perlmutter et al. (1999) received from their best fits values of Ω_Λ = 0.72 and ΩM = 0.28. Their data from the measured 42 high redshift Type Ia supernovae from the Supernova Cosmology Project and the data from 18 low redshift (z6 0.1) supernovae from the Calán/Tololo Supernova Survey are shown in Figure 2.5.

The effective rest frame B magnitude corrected for the width luminosity relation, mBis plot-ted as a function of redshift z. This is called an observed Hubble diagram. In the original Hubble diagram, the apparent magnitude is plotted as a function of redshift (Hubble 1929, 1936). Sandage (1956) called the "red-shift" speed of recession and plotted it as a function of apparent magnitude. The theoretical curves for mass density ΩM= 0, 1, and 2 and cosmo-logical constant Λ = 0 are shown as solid curves (from top to bottom), whereas the dashed curves show a range of flat cosmological models for ΩM = 0, 0.5, 1, and 1.5 and the total mass energy density ΩΛ+ ΩM = 1 with ΩΛ= _3HΛ2

0

as the contribution for vacuum energy. It can be seen that models with Ω_Λ= 0 are clearly ruled out. However, currently we lack the theoretical understanding of this value, ΩΛ≈ 70 %.

"For the discovery of the accelerating expansion of the Universe through observations of distant supernovae"17, Saul Perlmutter18, Brian P. Schmidt19, and Adam G. Riess19 were awarded with the Nobel Prize in Physics 2011.

17_{"The 2011 Nobel Prize in Physics - Press Release". Nobelprize.org. Nobel Media AB 2013.}

Web. 12-27-2013. <http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/press.html>

18_{The Supernova Cosmology Project}

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2.5 Definite departure from the canonical behavior

Despite of the achievements in cosmology that were achieved by using supernovae, the idea that all Type Ia supernovae are exactly the same is rather simplistic. The accuracy to deter-mine distances using Type Ia supernova is ≈ 10 %. Definite departures from the canonical behavior shows the much brighter SN 1991T (Filippenko et al. 1992b; Phillips et al. 1992; Wheeler & Benetti 2000). At maximum light it was 0.6 mag brighter in V than a typical Type Ia supernova (Wheeler & Benetti 2000) and shows Fe III but not the "classical" Si 6355 Å blend20 nor sulfur or calcium (Filippenko et al. 1992b). The other extreme is the subluminous SN 1991bg, which was at maximum brightness 1.6 mag subluminous in V and 2.5 mag subluminous in B (Filippenko et al. 1992a; Leibundgut et al. 1993; Turatto et al. 1996; Mazzali et al. 1997; Wheeler & Benetti 2000). Moreover, it shows no evidence for a secondary infrared peak in its light curve. These two supernovae are now eponymous for supernovae with these properties. They are generally called "peculiar". An other exception is the peculiar Type Ia supernova SN 2001ay. It was not overluminous in optical bands at max-imum light (MB= −19.19 and MV = −19.17 mag) but showed with ∆m15(B) = 0.58 ± 0.05 a brightness decline rate as three of the four overluminous super-Chandrasekhar mass can-didates (Krisciunas et al. 2011). This makes it the most slowly declining Type Ia supernova. Baron et al. (2012) showed that it consisted of 80 % carbon and that the unusual light curve decline can be explained within the framework of normal supernovae. As final exemption should be mentioned the Type Iax supernovae (Foley et al. 2013). It is characterized by the prototype SN 2002cx (Wood-Vasey et al. 2002b; Li et al. 2003). It was published by Li et al. (2003) as "The most peculiar known Type Ia supernova" with a premaximum spec-trum similar to SN 1991T, but a luminosity like that of SN 1991bg, expansion velocities of only 6 · 103km s−1, and many spectral peculiarities. Branch (2004), however, showed that SN 2002cx was a good example of a low photospheric velocity Type Ia supernova, a low lu-minosity SN 1991bg-like event that, in accordance with Marietta et al. (2000) is viewed right down a hole in the ejecta caused by the presence of the secondary star. Jha et al. (2006) con-sidered it as prototype of a new subclass, with spectral characteristics that may be consistent with recent pure deflagration models of Chandrasekhar mass thermonuclear supernovae. Ultimately, these observations were only possible with extensive monitoring programs and the discovery of a statistically relevant number of supernovae. As mentioned in Section 2.1, the supernova rate of a galaxy is rather small with a few per millennium.

This is recognized in the original IAU designation21. In the upper left corner of Figure 2.6 the total number of discovered supernovae per year of the last hundred years (1913 – July 2013) is shown.

The data are taken from the IAU Central Bureau for Astronomical Telegrams22. In the lower left corner of Figure 2.6, a cutout of the upper diagram until 1991 is shown, while the ordinate is scaled. Beginning in the 1990s, a larger number of supernovae were discovered. In the upper right corner of Figure 2.6 the supernovae that are certainly recognized as Type Ia

20_{Collective designation for Si II at 6347 Å and 6371 Å}

21_{The common name is formed by combining the prefix SN, the year of discovery and an upper case letter}

from A to Z. So, however, previously were only 26 per year possible.

22_{IAU Central Bureau for Astronomical Telegrams.}

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2.5 Definite departure from the canonical behavior 1 9 2 0 1 9 3 0 1 9 4 0 1 9 5 0 1 9 6 0 1 9 7 0 1 9 8 0 1 9 9 0 0 1 0 2 0 3 0 4 0 C o u n ts Y e a r 1 9 2 0 1 9 3 0 1 9 4 0 1 9 5 0 1 9 6 0 1 9 7 0 1 9 8 0 1 9 9 0 0 2 4 6 8 1 0 C o u n ts Y e a r 1 9 2 0 1 9 3 0 1 9 4 0 1 9 5 0 1 9 6 0 1 9 7 0 1 9 8 0 1 9 9 0 2 0 0 0 2 0 1 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 c e r t a i n l y r e c o g n i z e d T y p e I a s u p e r n o v a e C o u n ts Y e a r 1 9 2 0 1 9 3 0 1 9 4 0 1 9 5 0 1 9 6 0 1 9 7 0 1 9 8 0 1 9 9 0 2 0 0 0 2 0 1 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 T o t a l s u p e r n o v a e C o u n ts Y e a r

Figure 2.6: Graphical representation of the number of discovered supernovae per year of the last hundred years (1913 – July 2013). In the upper left corner the total number of discovered supernovae per year is shown as bar graph. In the upper right corner the part of supernovae that is certainly recognized as Type Ia supernovae is shown. In the lower part of the diagram cutouts of the upper diagrams until 1991 are shown with a scaled ordinate. The data are taken from the IAU Central Bureau for Astronomical Telegrams. See text for details.

supernovae is shown. In the lower right corner a cutout of the upper diagram until 1991 with a scaled ordinate is shown.

Especially those supernovae that differ somewhat from the canonical behavior are very inter-esting. They allow for model testing, that is the determination of to what extent the models reproduce a process quantitatively. One of the questions that remains is, whether the mech-anisms of these supernovae are really different, or depart only from the ordinary and can finally be explained in the framework of our theoretical understanding.

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Chapter 3 Parameter sensitivity of Type Ia

supernovae simulations from a

parameter estimation perspective

Parameter estimation is a term for mathematical procedures to obtain reasonable values for model parameters based on data. The identification and estimation of parameters is an in-verse problem, and thus belongs to a class of mathematical problems that are commonly hard to handle and not always uniquely solvable. However, compared to the classical forward problem, mathematical parameter estimation has many advantages, e.g., several parameters can be determined simultaneously, with distinct error ranges.

On the one hand there are often sets of noisy experimental and observational data. On the other hand there are theoretical models that are supposed to correctly reproduce a given process quantitatively. Although the first fine analysis of a star1 was performed more than 70 years ago (Unsöld 1942a,b,c, 1944) and much progress has been achieved since, an or-derly mathematical corresponding error analysis does not exist. Even today the process of parameter estimation often involves parameter tuning by hand or comparing the observed spectra with a grid of synthetic flux spectra within relatively wide parameter ranges. In as-tronomy, a well established technique is the minimum distance method (Bailer-Jones 2002). A special case of this method is the well known χ2minimization. This approach goes back to A.-M. Legendre2and J. C. F. Gauss3. J.-L. Lagrange introduced algebraic equations as constraints into the minimization problem4. Later inequalities were introduced as constraints (Karush 1939; Kuhn & Tucker 1951), nowadays we know it as Karush-Kuhn-Tucker (KKT) conditions. They form the generalized method of Lagrange multipliers.

Recently, Bock (1981, 1987) developed a boundary value problem approach and applied it, e.g., to the denitrogenation of pyridine5, where the solution of the boundary value problem and the minimization are performed simultaneously. He used a system of ordinary differen-tial equations as a model for the reaction kinetics.

In astronomy, classically, the problem of determination on the basis of a model grid and unconstrained minimization would in most cases require the calculation of 10n atmosphere

1_{Tau Scorpii}

2_{A. M. Legendre, 1805, Nouvelles méthodes pour la détermination des orbites des comètes}

3_{Carolo Friderico Gavss, 1809, Theoria motvs corporvm coelestivm in sectionibvs conicis solem ambientivm}

4_{J.-L. Lagrange, 1788, Méchanique analitique; par M. de la Grange, [. . .]}

5_C

5H5N, a basic heterocyclic organic compound that can be recycled into the valuable substances ammonia

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Chapter 3 Parameter sensitivity of Type Ia supernovae simulations from a parameter estimation perspective

models with n being the number of free parameters (cf. Wehrse & Rosenau 1997) and es-pecially for light curves of Type Ia supernovae, Section 5.3). Despite the effort to calculate and store this large number of synthetic spectra this method is much faster than the empiri-cal method of changing parameters until the empiri-calculated spectrum converges to the observed one. Even for experienced practitioners, the latter approach takes longer and can be at most randomly objective. Reliable error estimations are hardly available at all. The usual method to estimate the accuracy by comparing analysis from different authors does not necessarily yield good results. The discussion on the first carbon dwarf star (Gass et al. 1988) and on the iron abundance of Vega (Sadakane 1990) illustrates the problem. A detailed discussion of errors in astronomy and particularly in spectroscopy is given in Wehrse (1991).

However, the method developed by Bock (1981, 1987) is now used in many fields, and we plan to apply it directly to the astrophysical problem of parameter sensitivity of Type Ia supernovae.

We describe the parameter estimation problem as a process, which can be modeled by sys-tems of partial differential equations.

From modeling, there are the state variables,

y(t, τ, λ ,P), (3.1)

y includes the states, i.e., dependent variables. The states of the model are the solutions of the governing equations, e.g., partial differential equations, such as the time dependent radiative transfer equation (cf. Section 4.4, Eq. (4.66)). The independent variables are, e.g., the time t, a local variable τi, and the wavelength λi. It is assumed that the object, for which the spectral analysis is carried out, is fully described by the parameter vector for the model

Pm= Te f f, g, a, ξ , . . . . (3.2)

It is noted, that the element abundances adepend on the number of considered elements. For the model equations we write in general form

F(y, y,. Pm) = 0, (3.3)

which contains differential operators, initial conditions, and boundary conditions. We assume that for givenP, F has a unique solution, y(t,τ,λPm).

Adapted to the measurable values, the model response is

h(t, λ , y,Ph) . (3.4)

The parameter vectorP is composed of the parameter vector of the model (Eq. (3.2)) and the parameter vector of observationPh

P =Pm

Ph

. (3.5)

From observations we have the measured values ηi j at time points tj, j = 1, . . . , Nt, and wavelengths λiwith known variances σ_i2, i = 1, . . . , Nλ.

We assume the data obey a nonlinear regression

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with given observation functions hi j and εi j stochastic errors, that are assumed to be inde-pendent and normally distributed,

εi j ∼N 0,σi2 . (3.7)

The εi j does not depend on the observation epoch.

In parameter estimation the partial differential equation system is treated as an implicit con-straint. The parameter estimation problem is given by the minimization of a sum of weighted residuals and can be written for the corresponding norm as (cf. Bock 1987),

min y,P = Nt

∑

j=1 Nλ

∑

i=1 ηi j− h tj, λi, y,P 2 σ_i2 . (3.8)

The minimizing of Eq. (3.8) is a maximum likelihood estimate. The obtained variables fulfill the model equations, subject to

F(y, y,. Pm) = 0. (3.9)

Since the measured values are random variables, there are also the estimated parameters. The sensitivities Ji j,k = d dPk 1 σi ηi j− h tj, λi, y,Ph = −1 σi ∂ h ∂ y ∂ y ∂Pk + ∂ h ∂Pk λ =λi t=tj (3.10)

form the entries of the JacobianJ ∈ RNt·Nλ×nP of the residuals with respect to the

param-eters. The number of the parameter is denoted by k.

FromJ , the variance-covariance matrix of the parameter estimate can be calculated by C ( ˆP) = J+ _J+T

= JTJ −1, (3.11)

where ˆP denotes the estimated parameters and J+the solution operator,

J+₌ _JT_J−1_JT_, _(3.12)

which is a generalized inverse ofJ .

J and C do not depend on the observational data. The diagonal elements of C can be interpreted as variances of the particular parameters. The off-diagonal elements are their covariances and describe their correlation.

In order to enforce model validity during the parameter estimation computations, constraints can be considered, e.g., box constraints to the parameters

Pl

k<Pk<Pku. (3.13)

Pk (cf. Eqs. (3.5) and (3.2)) may represent, e.g., element abundances, temperature, or the parametrized line scattering scattering fraction. The latter can be set for line scattering εline

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Chapter 3 Parameter sensitivity of Type Ia supernovae simulations from a parameter estimation perspective

to be in a range close to 0, which means that almost 100 % of photons are scattered via line scattering, and 1, which means that no photons are scattered,

εline∈ ]0, 1] . (3.14)

Normally εline is computed and not used as a parameter. For test and special cases it can be used as a parameter (cf. Sections 5.5 and 5.6).

The parameters are obtained by least squares minimization constrained by a boundary value problem.

An approximation for the 100 · (1 − α)% confidence region of the estimated parameters ˆP is described by the linearized confidence range

G(α, ˆP) = nP + δPˆ δ P T_JT_{J δP 6 γ}2_(α)o _(3.15) = nP + δPˆ δ P T_C−1 δP 6 γ2(α) o = ˆ P + δP C −1 2_δP 2 26 γ 2_(α) ,

k k2₂ is the square of the Euclidean norm, and γ(α) the quantile for the value α of the chosen distribution. In this case, it is the χ2distribution6.

The confidence ellipsoid of the parameter estimate is described byC−1.

6_{Because we assume that we know the variances σ}2

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Chapter 4 Modeling spectra and light curves of

Type Ia supernovae with

PHOENIX

4.1 Radiative transfer

4.1.1 Radiative transfer in general

Radiative transfer describes the energy transfer in the form of electromagnetic radiation. If crossing material, radiation is influenced by absorption, emission, and scattering. Many of these effects can not be explained classically, but must be considered quantum mechanically. Lord Rayleigh (1871) was the first who considered scattering effects. He developed a math-ematical boundary value problem, with many assumptions, often not accurately stated and demonstrated that the sky is blue due to the stronger scattering of light at shorter wavelengths. Next, Mie (1908) extended this theory by looking at additional absorption. Mathematically, it is again a boundary value problem and now even more complex. The first person who considered multiple scattering was Schuster (1903, 1905), more than a century ago. He as-sumed that the light passes through a plane in two directions, both perpendicular to the plane. Mathematically, Schuster’s differential equations depend only on the depth of penetration. Schwarzschild (1914) integrated over all directions in the forward and in the backward hemi-sphere, so that his equations in addition to the depth of penetration take into account the an-gle of incidence θ , i.e., for the forward direction 0 < θ 61₂π and for the backward direction 1

2π < θ6 π. His equations are more general than in the original formulation.

Today, we can derive the radiative transfer equation in several ways, e.g., from the Boltzmann equation by linearization (Oxenius 1986), from quantum field theory (Sapar 1978; Landi Degl’Innocenti 1996), by means of a stochastic model (von Waldenfels 2009; von Waldenfels et al. 2011), or, as shortly introduced here, phenomenologically. Wehrse & Kalkofen (2006) summarized the progress in radiative transfer together since 1985.

Radiative transfer is important in all areas where light is used as diagnostic or modeling tool, e.g., in medicine, combustion engine design, environmental and plasma physics, and in as-trophysics, where photons are by far the most important source of information. Therefore, especially for an astronomer, the modeling of the formation of the radiation field and its propagation is of particular interest. The radiation field links microscopic effects from the interaction of photons with matter to the observable macroscopic effects. Unfortunately, the computational effort is enormous, despite the progress in recent years in computer technol-ogy. In addition to the mathematical and numerical challenges, such as the strong variability (a few dex) of the extinction coefficient entering the radiative transfer equation and the

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enor-Chapter 4 Modeling spectra and light curves of Type Ia supernovae with PHOENIX

mous number of atomic lines that have to be included. Nevertheless, in astrophysics, it is one of the most important equations that allows to examine the conditions in the Universe, a stellar atmosphere, or, as here in this work, the atmosphere of a Type Ia supernova.

In this chapter, the description of the radiation field and the basics of radiative transfer are introduced and briefly discussed.

4.1.2 The specific intensity and its moments

dS s Iv n dw Θ

Figure 4.1: Geometric basis for the definition of the specific intensity Iν (based on Unsöld (1968) and Mihalas (1978), with modifications). See text for details.

The radiation field of unpolarized or natural light is described by the specific intensity I_ν(r, s, ν,t), which is in general a function of the position in space r, the propagation direc-tion n = (cos φ sin θ , sin φ sin θ , cos θ )t, frequency ν, and the time variable t. The subscript refers to a normalized frequency interval (cf. Unsöld 1968). As can be seen in Figure 4.1, θ is the angle between the direction of the propagation of radiation n and the outward normal s to the plane dS. The radiation is filling a truncated cone dω = sin θ dθ dφ around the direc-tion n, through a point P of dS in a time interval dt, and a frequency range ν . . . ν + dν. This amount of energy dE is in astrophysics traditionally written as,

dE = Iν(r, n, ν,t) cos θ dσ dt dω dν, (4.1)

and goes back to Unsöld (1938) and Chandrasekhar (1950).

Alternatively, dealing with light in the particle picture, the specific intensity can be written by its relation to the photon distribution function φ (r, n, p,t) normalized in a particular way,

I_ν(r, n, ν,t) = h 4

ν3

c2 φ (r, n, p, t), (4.2)

where h is the Planck constant1, c the speed of light2, and p = hν_c n the photon momentum (Oxenius 1986).

1_h_{= 62606957(29) · 10}−34_{J s}

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4.1 Radiative transfer

In the wave picture, we have to use the more complex definition from quantum field theory (cf. Wehrse & Kalkofen 2006; Kanschat et al. 2009). The radiation field is given in terms of the electric and magnetic fields E(r,t) and B(r,t). Since these two fields are linked to each other through Maxwell’s equations and special relativity, it is sufficient to use only one of the variables. Here we have chosen the electric field E(r,t), which is in a plane perpendicular to the propagation direction and oscillates in phase with B(r,t). It is given by Maxwell’s equation E = −∇Φ −∂ A

∂ t , which is reformulated in terms of (magnetic) vector potential A and (electric) scalar potential Φ (Stenflo 1994),

E(r,t) = −∂ A ∂ t = E

+_{(r,t) + E}−_(r,t), _(4.3)

with the two field vectors E+(r,t) and E−(r,t) at the space-time point (r,t) (Mandel & Wolf 1995; Vogel & Welsch 2006),

E+(r,t) = √1

V _k,σ

∑

l(ω)ˆakσekσexp(i(k · r − ωt)), (4.4)

E−(r,t) = E+(r,t)†, (4.5)

where V is the quantization volume, l(ω) a function of ω, e.g., q

}

2ω0 for the vector

po-tential, } the Dirac constant3, 0 the vacuum permittivity4, ˆakσ an operator, ekσ is an unit polarization vector orthogonal to the wave vector k = kn, and ω the corresponding frequency. We assume that the field is in a mixed state and summing over all final states hψj| and there-after over all initial states hψi|,

I(r,t) =

_∑

ψj

∑

ψi hψi| E−(r,t) |ψji hψj| E+(r,t) |ψii =

_∑

ψi hψi| E−(r,t)E+(r,t) |ψii = Tr ˆρ (r, t)E−(r,t)E+(r,t) . (4.6) The details with the density operator ˆρ (r, t) defined by

ˆ

ρ (r, t) =

_∑

i

P_i|ψii hψi| (4.7)

and characterizes an ensemble of states hψi| where Pi is the probability that a randomly selected particle of the ensemble is in the state, which is described by the wave function ψi (Cohen-Tannoudji et al. 1999).

The Stokes vector may be expressed as

I =     I Q U V     =       G(1)₁₁ + G(1)₂₂ G(1)₁₂ + G(1)₂₁ i G(1)₁₂ − G(1)₂₁ G(1)₁₁ − G(1)₂₂       , (4.8) 3 } =2πh = 1.054571726(47) · 10−34J s 4 0= 8.854187817[. . .] · 10−12F m−1

Parameter sensitivity of synthetic spectra and light curves of Type Ia supernovae